Application to Dose Calculations


Since the Boltzmann equation includes collisions between particles, it is possible to estimate the energy delivered to the tissue and thus the dose in radiotherapy. Greater efficiency is achieved by working with particle distributions (photons, electrons and positrons) rather than individual particles as in the case of Monte Carlo. The introduction of cells using the Lattice Boltzmann Method (LBM) should also allow the possibility of choosing the precision according to the computational resources that are made available. This means that the algorithm should be able to pass continuously from a system equivalent to Pencil Beam, Convolution to Monte Carlo.


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Thermal Photons


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For the case in which they are considered uniformly distributed thermal photons their number per cell will be according to the distribution of Bose-Einstein

\displaystyle\frac{1}{e^{\hbar\omega/kT} -1}

where \hbar is the Planck constant divided by 2\pi, \omega is the angular velocity, k the Boltzmann constant, and T the temperature.

If the flow is isotropic it will be necessary that the $ m components will be equal and therefore:


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Caso Electrones


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In the case of the electrons it is necessary that the distribution in equilibrium is the one of Fermi-Direc so in the situation of equilibrium the distribution will have to be of the form

$f^{eq}_i=\displaystyle\frac{1}{e^{\beta (m_ev_i^2/2-\mu)}+1}$

Otherwise the possible scatterings correspond to

- Absorption

- Elastic collision

- Electron-electron collision

- Excitation and deexicitation

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Scattering that contributes (in) or describes the abandonment of particles (out) can be plotted as follows:

It should be noted that the term collision:

- integrates on all external speeds to those of volume

- includes the likelihood of both speeds leading to scattering simultaneously

- the relative velocity multiplied by the total effective section represents the flow of particles towards the target

The latter can be shown in a simple way through

\Delta v\sigma\sim\displaystyle\frac{dX}{dt}S\sim \displaystyle\frac{dV}{dt}\sim J

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In case the particles collide, the distribution function f(\vec{x},\vec{v},t) variert und\\n\\n

$\displaystyle\frac{df}{dt}\neq 0$

Collisions cause particles of neighboring cells to undergo a collision that takes them to the cell under consideration and particles within the cell being expelled. The first leads to an increase of f_{in} particles and the second to a f_{out} time loss \tau. Thus the Boltzmann transport equation with collisions can be written as


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Collisions leaving the cell


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In the case that they leave the cell it is considered


Integrating on one of the speeds that initiate the collision and both resulting since the other is the contribution to the local distribution function

$\displaystyle\frac{1}{\tau}f_{out}(\vec{v})=\displaystyle\int d\vec{v}_1d\vec{v}_12d\vec{v}_22f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v},t)|\vec{v}-\vec{v}_1|\sigma(\vec{v},\vec{v}_1\rightarrow\vec{v}_12,\vec{v}_22)$

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Collisions that contribute


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In the case of contributions to the cell, consider


Integrating on the speeds that initiate the collision and one of the resulting ones since the other is the contribution to the local distribution function

$\displaystyle\frac{1}{\tau}f_{in}(\vec{v})=\displaystyle\int d\vec{v}_1d\vec{v}_2d\vec{v}_12f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v}_2,t)|\vec{v}_2-\vec{v}_1|\sigma(\vec{v}_1,\vec{v}_2\rightarrow\vec{v}_12,\vec{v})$

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